A note on recursive maximum likelihood for autoregressive modeling

Abstract

Abstract-In this paper, we rederive recursive maximum likelihood (RML) for an autoregressive (AR) time series using the Levinson decomposition. This decomposition produces a recursive update of the likelihood function for the AR parameters in terms of the reflection coefficients, prediction error variances, and forward and backward prediction errors. A fast algorithm for this recursive update is presented and compared with the recursive updates of the Burg algorithm. The comparison clarifies the connection between Burg's algorithm and RML.where I is the identity matrix, and J is the exchange matrix

Nonlinear maximum likelihood estimation of autoregressive time series

“…Burg [4] LeRoux-Gueguen [13] Vis-Scharf [19] Morf et al [16] Friedlander et at. [9] Dugre et al [7] Demeure-Scharf [6] Kailath et al [11] Gohberg-Semencul NA NA Kay [12] Box-Jenkins [3] Mcwhorter-Scharf (22)…”

“…From (19) we write Rp(Y)a +~(7"2Qpad = (7"28 Q;I Rp(Y)a +~ad(7"2 = (7"2Q;1{) = aO"2. (20) The matrix Qp is the (p + 1) x (p + 1) northwest block of the matrix Q defined in the Gohberg-Semencul formula of (1).…”

“…We classify this work as recursive maximum likelihood (RML), with a Gohberg-Semencul formula for the inverse correlation matrix. The work of Vis and Scharf [19] is classified as RML with a Levinson formula for the representation of R -1 and the order increasing AR models. It clarifies the connection between Kay's work on RML and Burg's work on RLP, and completes the classification of the literature in the third column of the table.…”